Additional Examples of sample spaces and events
Example 1: In the experiment of tossing a single regular six-sided die, there are six possible outcomes.
Thus the sample space is
Ω={
,
,
,
,
,
}.
This may also be depicted simply as Ω={1, 2, 3, 4, 5, 6}. Let A be the event that even number of spots
turn up, i.e.
A={
,
,
}.
Or A={2,4,6}. A is an event; that is it consists of a subset of elementary outcomes in Ω. Since in this
example the sample space is finite, the event space, sometimes denoted by A. , consists of all subsets of
Ω. Thus, there are 2 6 =64 events, of which only 6 are elementary, in A. , including both φ and Ω.
Example 2: Toss a penny, nickel, and a dime simultaneously, not whether heads (H) or tails (T ) is up.
There are eight possible outcomes of this experiment; thus Ω={(H,H,H), (H,H,T), (H,T,H), (T,H,H), (H,T,T),
(T,H,T),(T,T,H), (T,T,T)}. We are using the first position of the 3-tuple (∙,∙,∙) to record the outcome of the
penny, the second position to record the outcome of the nickel, and the third position to record the
outcome of the dime. Let Ai = {exactly i heads}; i = 0,1, 2, 3. For each , Ai is an event. Note that A0 and
A3 are each elementary events. All subsets of Ω are events and there are
of them.
Example 3: This experiment is to record the number of traffic deaths in the state of Iowa this year. Any
nonnegative number is a conceivable outcome of this experiment; so the Ω={0, 1, 2, …}. The event A that
the number traffic deaths is less than 500 is A={0, 1, 2, …,499}. The event Ai = { exactly i deaths}, for
i=0, 1, 2,…, is an elementary outcome. The sample space is infinite but countable.
Example 4: Select a light bulb of a fixed wattage from a warehouse and determine its lifetime. Any
nonnegative number is a conceivable outcome of this experiment; so Ω={ x : x ≥ 0 }. For this sample
space not all subsets are events; however, any subset that can be exhibited will be an event. For
example, let A={bulb burns for at least k hours, but burns out before m hours} = { x : k ≤ x ≤ m }; then A
is an event for any 0 ≤ k < m .
Example 5: Consider a random experiment which consists of counting the number of times it rains and
the total number of inches of rainfall for the month of May in Ames. The sample space could then be
represented by Ω={ (i, x ) : i = 0,1,2,; x ≥ 0} . For example, ω = (7,2.251) is an elementary outcome in Ω.
A={ (i, x ) : i = 5,,10; x ≥ 3} is an example of an event. (i, x ) : i = 0,1,2,; x ≥ 0} . (i, x ) : i = 0,1,2,; x ≥ 0}